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Parents, here is an overview of the curriculum for next year’s math classes for children around early second grade — Thursday morning in Alameda and Tuesday morning in Concord. CURRICULUM — MIQUON The Concord class will begin in the fall at about halfway through the Miquon Red Book, and we will choose topics as needed for the children to finish up the Red Book, then begin the Blue. The Alameda section, having stared in January of this year, is ready to begin the Blue Book in September. To help you understand the Miquon series, I’ve reproduced three representative pages from the Red Book, with comments. Because Miquon organizes their workbooks into more or less parallel topics, more advanced students can work with similar topics in the Blue book while others work in the Red Book. This works because what I teach in the lessons is problem-solving, which isn’t taught in any of the Miquon books, nor any other standard curriculum. You need live lessons for that. But the parts complement each other: problem-solving in our weekly lessons together plus skill reinforcement during the week as homework. red book fractions This is the last page of Miquon’s 24 page introduction to fractions. Miquon (and the Waldorf curriculum) is unusual in teaching all four operations in first grade, plus fractions, but I think this is correct. Children should get this broad conceptual understanding from the beginning. red book division The last page of the division unit. Children can solve these problems with Cuisenaire Rods or a novel abacus invented by the RightStart Math. Additionally, now that they know the meaning of all the operations, the children can solve word problems. With my Alameda class recently, for example: The baker baked sixteen pies, but after she set them out to cool, a rascally boy took two of the pies and ate them. Half of the remaining pies were lemon, half were cherry. How many lemon pies were there? Children solve this three ways: arranging manipulatives to tell the story, with a drawing, and finally with a math sentence. red book multiplication This page demonstrates one of the reasons I like the Miquon series. These two sets of questions are quietly about the distributive property and about average, without cluttering things by prematurely mentioning those terms. This is math the way a mathematician sees it, and it is intellectually beautiful. In class, the children translate the ideas behind these exercises into words and then into hand gestures. CURRICULUM – RIGHT START I encourage parents to use a second curriculum alongside the Miquon books. RightStart Math is a curriculum of 300 math games, designed by a mathematician, and strongly influenced by Montessori methods. They bundle a kit with several decks of cards, game instructions, and an innovative abacus. The games appeal to young children and they offer a good complement to the Miquon workbooks. As good as Miquon is, it is even better for the child to get their nose out of a workbook now and then and do math in a more visceral game format. TIME In September, I suspect the children would only be up for lessons that are 1 1/2 hours, but I would like to ramp them up to being able, by January, to lessons that are two hours long. I’ve been successful at keeping childrens’ enthusiasm for 1 1/2 hours without a break, but it all depends on the group. For a longer class, they would need a small break. SCOPE/SEGUE The Miquon series is paced at one workbook per semester. We will stay with Miquon through Red, Blue, Green, Yellow, and at least part of Purple, and then segue in about two years to the Beast Academy series, published by The Art of Problem-Solving. This is the deepest and best-written math curriculum that I’ve found for grades 3-5. I can talk at length about it and show you some of their books, but that math is two years away! In the mean time, if you have more immediate questions about next year’s math, please email or phone me. hducharme@gmail.com , 510 417 5736.

When I mentioned yesterday to the SuperGirl 5 Math Group that the next unit coming up in their school curriculum will be adding fractions, there was a huge chorus of groans, and they took a minute to vent. They all complained that “Adding fractions is soooo boring.” and they seemed to use ‘boring’ in three different senses: • This is not challenging us. We’ve already done this before. • This is pointless. There is no good reason to be adding fractions anyway. • This is treating us like little kids, insulting our intelligence. We’ve been doing fractions since second grade. Of course, whether these perceptions are true or not matters less than that they are indeed the girls’ perceptions. There’s also a paradox here, that they may be bored with adding fractions but I know from SGM Summer Camp that they are shaky on even understanding what fractions mean, and in a well-sequenced curriculum, understanding what fractions mean comes before adding them up. But maybe it would be better to not to have a well-sequenced curriculum. As luck would have it, I had planned two activities to teach the meaning of fractions, but mixed up were a bunch of other topics they won’t see till sixth grade or so. The first was a poker-like card game. One round, the winner was the person who had the best cards to fulfill 1 / (x + 2y) = 0 (Winner is she who comes closest to 0) This led to the tricky strategy that if you want the fraction to be as small as possible, then the denominator, (x + 2y), should be as big as possible — which I could only explain by reaching for those boring pizza slices they shared back in second grade to demonstrate the meaning of fractions. In another round, three girls dropped out of the bidding because they claimed they had poor hands for (x + y) / 2 (x + y) = 1/2 (Winner is she who comes closest to 1/2) What they didn’t realize is that any two cards substituting for x and y will always give 1/2 for the whole expression. Everybody is a winner! I connected this to the boring fourth grade idea of equivalent fractions. In the second activity, each girl had a number taped to her back and had to deduce what that number was. The only clue was that if each added their hidden number to the number on the back of some other girl, the two would add up to 1, so they had to figure out who was paired with whom. In one round, Laila deduced, for example, that 200% must equal 2, since that person had to be paired with someone who had −1 on her back, and 2 + −1 = 1. Although none of them could have solved any whole round individually, they could solve each round together in about ten minutes. At this stage in their curriculum, they haven’t been officially taught percentages, but they’ve certainly seen then around, and maybe the girls are right, maybe it would insult their intelligence to pretend that they can’t do any problem-solving with percentages. And the same with the several other topics that I had jumbled into the mix, including all of the algebra topics that we discovered in our poker game. The fun of problem-solving isn’t just for advanced students, and this group is more or less average, “at grade level”. We still ran into such misconceptions as 1/8 = .8, or 1/0 = 1, each of which had to be carefully debunked, again. But there appeared to be a lot of excitement and a lot of messy learning yesterday, and perhaps that was more important than the sometimes bumpy ride. Students have taught me something similar in chess class. There is a months-long stretch after learning how the pieces move but before learning to play a whole game, and years ago I designed a curriculum to cover that stretch. But students rebelled (boys especially), saying in effect, “I don’t want you to teach me all these clever ideas. I would rather have the excitement of playing many games, losing many games, and slowly bumping into those same edeas. Then will be the right time to teach me.” The style is more ad hoc and opportunistic, like yesterday when we figured out that Naomi would have edged Meliya by .1 in a poker round if she had only played her cards right. After that motivating loss, I know she’ll nail it the next time around. But in chess, it took me about two years to ease up on my carefully planned curriculum, because I can be quite stubborn when I know I am right.

We began with a continuation from our last lesson: “What is the highest number that you can name, can write, and would know how to count up to?” When we left off that highest number was 12, 999. After another fifteen minutes of new answers and discussion, in which I kept responding, “…And what would be one higher than that number?” they rested at a new summit, 99,999. After that, the abyss. We also reviewed number circles from last lesson. We used the number circle to chant the 2’s times tables. Then we moved to today’s lesson. The scroll suspended from the ceiling shows the pattern of doubled numbers: 1 , 2 2 , 4 3 , 6 … 19 , 38 20 , 40 . We noticed various patterns in the chart, and talked about how you could use those patterns to try to predict, “What would be double 14?” etc. You may be wondering what is an accordion doing there. I wanted to give them a calm, even serene experience of doubling. So I played a calming Phillip Glass piece as I questioned and they answered on beat, “18?” (beat 2 .. beat 3 .. beat 4..) “36” …. I had asked the girls to bring meditation cushions. Some took it quite seriously and chanted their doublings with upright posture. Then break. After break, we observed doubling through folding newspapers, 1, 2, 4, 8, 16. When they returned to our room, the scroll had been changed. It had been taken down and a new one put in its place, but this one was blank on both sides. They were surprised. We ended with a final doubling practice with only the blank scroll to focus their attention. The very last problem, accompanied by the Glass, was “0?” (beat 2 .. beat 3 .. beat 4..) “0.”

The Second Grade SuperGirl Math Class is currently full. Who: 8 Girls When: alternate Mondays, 4:00 – 5:30 Where: parent houses in Berkeley Why: challenge + review, to help girls to be excited about math, and to be strong in math How much: $150 for five meetings through December A second session will begin in January. Registration will open in early December. From the SuperGirl Math Camp over the summer: Well first off let me say it’s been great, thank you, and Shira went in one day from saying “I don’t like math” to saying “camp was great”. What can I say, you’re a gem! … I’d love to have it continue during the school year. — Tamar, parent I am very impressed with your commitment to this camp, and Naima is having a great time! … I would like to continue something with you during the school year …. Thanks again for making this so engaging for the girls!

Some daily highlights from our Camp In measuring the body proportions of some cartoon characters using rubber bands, we had to fold some paper. (It’s a long story, but your girls can explain.) In folding our papers, Laila had an interesting question. “The top paper is folded in fourths, and the bottom paper is folded in fourths, but since the top is way smaller than the bottom, maybe I should call it eights, or something else?” They discussed this for several minutes, and I then summarized the discussion by saying, “Maybe there is no such thing as ‘one fourth’, but maybe it always has to be one fourth of something.” Using Cuisenaire Rods, we made models of various halves, thirds, and fourths, and I asked the girls to arrange their answers in a way that was both beautiful and showed the logic of the math. This team was obsessed with the idea of the fractions portraying a fireworks, and I went along with that. We discussed how that idea might dictate the placement of the various fractions, and they came up with an imaginative answer. Enough photos of papers and gizmos. This is a candid of Meliya in pure thought. I showed the girls a way of visualizing multiplication, then we practiced on some problems such as 13 x 14 = ? Although we started solving these using wooden gizmos, the room became eerily silent when they solved using nothing but their imaginations. The point of this was not to learn yet another way of multiplying, but to demonstrate that multiplication can be pictured. That is a valuable lesson that they will use this year, although these particular problems are harder than what they will see in school. That picture of multiplication is directly related to the “lattice method” that they’ve already learned in school, although the students don’t realize that yet. DAY TWO: Math Field Trip I can’t believe that I hadn’t noticed this before, despite several previous visits to the exhibit. It looks at first glance as if each of the smaller rectangles in this work is 1/20 of the whole, but I was startled to notice that the rectangles are subtly different. Why? What’s going on? Wooden rulers appeared, and 6/9 of the students rushed to measure the rectangles. Figuring out ratio of length:width by subdividing. Sophie was the first to start doing this intuitively, but when I started referring to this as “Sophie’s trick”, there were several protestations of “I did it too!” DAY THREE 2 cups make a pint 2 pints make a quart 2 quarts make a half-gallon 2 half-gallons make a gallon Although this list is conventional for us, if you think about it, there is something odd with it, something that breaks the pattern. Our language has lost the older term for “half-gallon”, which used to be called “pottle”. Our early folk measurement system was elegantly based on the powers of 2, and the units continued far less than a cup and far greater than a gallon: …. 2 jills make a cup, …. 2 gallons make a peck, 2 pecks make a pail, …. I used this today as an introduction to thinking multiplicatively instead of additively. In other words, what is the next number — 2, 4,… ? Fourth graders will say that the pattern continues “2, 4, 6, 8…” and while that is true enough, fifth graders should start to see that there is another possibility, “2, 4, 8, 16,…” In the adult word, the second answer is far more important. In the photo above, students are solving, “How many gallons in a hogshead?” Ten minutes later, the answer is 256. Along the way, we again bumped into reciprocals. “If 16 cups make one gallon, how many gallons make one cup?” We had had a variation of this problem on Monday, when Stella complained, “This is making my head explode.” Although I haven’t formally taught reciprocals since then, her head is doing a lot better now. She raised her hand to answer the question, and then matter-of-factly said the correct answer. By Friday, I predict she would answer the question with some adolescent impatience, as in how could I be so dumb as to even ask her the question? Amanda and Leila solved the problem in a surprising way, although the fact that I call it “surprising” may just mean that I don’t understand how they think. It also meant that they were very happy. When Hannah and Nico (not pictured, sorry) worked on the problem, they realized the answer would be 16 x 16, and Hannah asked for paper and pencil to figure that out. But I reminded them, “Wait a second — we just did something like that the other day. Can’t you just solve this by imagining the picture instead of always running to paper and pencil?” And so they did. We ended the morning by developing a visual understanding of the four basic operations with whole numbers, and then extending that visual understanding to fractions. FIN

This inaugural demonstration camp was the “proof of concept”, and the concept seems to work: a girls-only math camp, focusing on what they’ll see come the new school year. For the next SuperGirl Math Camp coming up August 5-9 (also for fifth graders), we have nine girls signed up, one slot remaining. I promised the girls that we would begin and end end each day with “Weird”, an unusual application of fractions. On our first morning, we studied the proportions of Bart Simpson and other characters using marked-up rubber bands. You can see that Bart’ head measures a little more than a third of his overall height. We did two experiments applying ratios to cooking. One with orange juice: can the girls taste the difference between a 1:3 concentrate:water and a 2:5 ratio? Our second ratio experiment, trying different ratios of butter:sugar:flour for cookies. We also added various toppings. At first they referred to the cookies by their toppings (“the one with almonds and vanilla”) but they came to realize that toppings were superficial, and that the structure of the cookie was determined by the butter:sugar:flour ratio. Missing, for example, is a photograph of the inedible greasy cookie blob because the ratio of butter to flour was way too high. In the middle of each morning, we had two sessions of “Practice.” Here, the girls took 45 minutes to make representations of halves, thirds, and fourths. They are using Cuisenaire Rods, an elegant math manipulative that I’ve used to teach everything from kindergarten math to algebra. In the course of their work, we cleared up several misunderstandings they had about the nature of fractions. One of our two field trips was to the Hazel Wolf in the Brower Center, Berkeley, where there is an interesting exhibit on relationships and sociability — a relevant topic for fifth grade girls. In the photo are two maps of race by Census tract in Alameda County, the usual way you or I would probably represent it on the paper, and a very different way of visualizing it in the artwork. Both are true; both are very accurate. The girls stayed with this over a half hour to puzzle through the differences. The two sketches on the right are of a landscape in the exhibit, showing the proportions of rock, sand, and sky. But in her first drawing, the student saw the proportions inaccurately — that happens to most of us when we encounter extreme proportions; our mind doesn’t believe what our eyes see and we tend to see proportions as less extreme. The second is her corrected version. [And I made a mistake by not insisting that her second sketch be the same overall size.] We are fortunate to have camp in a home rather than in an institution, and we used the moods of three contrasting rooms for our daily Weird, Practice, and Talking Time (plus a fourth, the kitchen for our experiments.) During Talking Time, I asked questions that they answered in their math diaries and then shared aloud: what are your past experiences with math? when (in any field) have you taught yourself something new? And we continued the theme of “relationships.” Since we’ll be meeting twice a month during the school year, we discussed how we can support each other and work together as a team. During our Tuesday Math Walk, I posed a problem: “How much money does the City of Berkeley collect on this block in parking meter fees each day?” It took them 2 minutes, 51 seconds to solve, and I saw so many good morals from the process that I decided to go deeper into it. On Wednesday, after much discussion and several rehearsals, we shot a video reenactment of solving the parking meter problem. Then on Thursday, an unexpected payoff: overnight, one student spontaneously made up her own problem of a similar sort, “how many times does the gate in front of my house swing back and forth in a year?” She is starting to see her world differently! In the end, more important than giving the teacher the right answer is asking yourself interesting questions.

In this post, a description of our first three problems in compass constructions, there are no pictures of happy children playing with math, but I will try to imply their happy minds playing with math by describing the challenges I gave to the children and how they responded. You may have seen constructions in high school geometry, but instead of being taught by rote — as I was — the topic is presented as problem-solving challenges for fourth and fifth graders. The chance to dig in and pursue problem-solving, through mistakes and detours and dead ends and byways, makes all the difference, turning the topic from rote learning to an adventure. First challenge: Can you make this? Here is a student solution. Most of the students were able to get this within a half hour; a few figured it out instantly. For some, they first needed practice in manipulating the compasses before they could draw this. A few children drew something like the drawing below, which I praised in class as “a wonderfully wrong answer.” I’m not referring to the four petals on this flower rather than the six I drew for them. More important is that the student isn’t playing the construction game the way Euclid would want. In order to draw this, the student had to eyeball where to put the compass point, and that’s not allowed. Next class, I will challenge the students to figure out how to do this alternate construction, which is beautiful in its own way, but according to Euclid’s rules. Second challenge: draw an equilateral triangle. Most students tried to do something like this, a little measuring, a little eyeballing, both of which aren’t allowed and won’t get the job done. After a student showed me this solution, I asked her if she could use her compass to measure if all three sides are truly equal. She did — you can see the little arcs she drew — and found that it’s close, but not quite correct. Since they had done the first six-petal challenge correctly, I asked the students to study that and see if that could help them with the triangle problem. Could they see the implicit equilateral triangle? When some students excitedly told me they could see the triangle, even though it was not actually drawn, I told them that the are either 1) crazy or 2) wonderful mathematicians. Third challenge: find the midpoint of a line segment. Here is a clever incorrect answer, in which the student essentially started with the answer, a circle plus a diameter drawn through the center, already dividing it perfectly. A few other students got an iterative solution: they did trial bisections by eye, and kept adjusting the compass more and more finely until it appeared to be correct. Very clever, very advanced, but not allowed by Euclid. A couple students got the correct answer (that I was looking for) pretty quickly, but others want to figure it out for themselves, and aren’t interested in merely copying answers. The challenge for me as teacher is how to frame this problem in a way that can help the students to solve it. Next lesson, I will try to elicit from them that the solution is symmetrical, and even though the drawing looks very simple, being symmetrical is a big deal. It might suggest to some that we should make our constructions symmetrical if we hope get the same solution.. I think that should be enough to get many of them over the hump, but I really don’t know. We’ll all find out next class!

I met recently with half a dozen students of color at Jefferson Elementary, whose teachers had encouraged them to try out the Math Circle class. The idea was for me to lead a demonstration class in which they could meet me and sample the math. Besides the math that we tried out, compass constructions, I also wanted to explain to the students why I was there. Since we’ll be studying sets in the upcoming Math Circle, I decided to explain using the sets of a Venn diagram. I began by asking the students to fill in the blank: “All of us [the students] are _________, but Mr. Ducharme is not ___________.” After a few more obvious answers, a student gave the answer I was looking for: “We’re all brown and black and yellow and different colors, and you’re not!” I introduced the term “people of color.” I then put a 2’x3’ piece of paper before the students and asked them to grab about 150 little plastic bears I had borrowed from kindergarten, to represent all the third, fourth, and fifth graders at the school. They had nostalgic memories of playing with the bears in kindergarten, but I warned them that this lesson would go from being funny and fun to surprisingly serious. I drew the first circle of the Venn diagram, and asked them to divide the bears into two groups: students of color and not students of color (i.e., white students.) They came up with about 110 students of color and about 40 white students. I asked Ms. Reed, one of their teachers who was working with us, to correct their division. The correct answer is just about the complement of what the students had: over two-thirds to three-fourths of the students are white, only a minority are students of color. I introduced the term “bias” and we discussed possible reasons that their answers were so biased. The students were already starting to get quiet, and now serious as I added the second intersecting circle: students who are strong enough to do Math Circle vs. students who are not. I defined the the terms, we went over the possible sets, now four of them, and again I asked the students to distribute the tokens accurately. This is their answer before Ms. Reed made a few smaller corrections. Finally I added the third circle, students who would actually sign up for Math Circle class at Jefferson vs. those who would not sign up. We reviewed the eight resultant sets, I asked students to distribute the tokens one more time, and we discussed their answers. Then I told them, “I don’t know for sure what will happen here, I can’t predict the future, but I can tell you the past, what has happened in fact in my last three Math Circle classes at three different schools.” Using numbers instead of tokens and only looking at the set of student who signed up for class — for I won’t guess about students I didn’t teach — this is the sobering result. The optimistic students had placed about six tokens for the intersection of {students of color + strong enough to do Math Circle + actually sign up}, for their school alone. The reality, as of my last three schools, was nearly zero. That, as I explained, was why I had volunteered to come for an hour to introduce myself and Math Circle to them. We discussed these results and then moved on to more innocent math of compass and straightedge constructions. When I left, four of the students, who appear strong enough to do challenging math, wanted very much to sign up for Math Circle. Addendum, four months later: Those four did sign up and stayed with it, as did thirteen others to make a class of seventeen. We had majority girls and about half students of color. It was a great class.

In General: Math Circles began a generation ago in Eastern Europe, originally for high school students. Their purpose isn’t to cover topics in the usual algebra through calculus sequence, but instead to instill passion for math by presenting advanced math topics that a student would typically not see, topics that hint at what math looks like to a mathematician. Or more radically, as I have told some of my students, “What you think math is, is not math.” In recent years, Math Circles have been attempted for younger children in elementary school. But at each level, the structure is the same: a carefully chosen problem is presented to the students, and they take it from there. They come up with ideas, they discuss, they argue, they bump forward. They are guided by the teacher, but they make their own discoveries, they are free to make their own mistakes. It may take us an hour or two or three more to work through a problem. One solution may suggest other problems, perhaps a conjecture, a generalization. We begin again. Done well, the process gives students a sense of excitement and wonderment for the beauty of higher mathematics. This Session: I teach three Math Circle classes at Jefferson, Berkeley Arts Magnet, and Malcolm X schools. We will cover four new topics: – constructions with compass and straightedge, hopefully rediscovering the beginning of Euclid’s geometry – some math behind Braille + graphing our learning – making and solving Soma Cube puzzles – playing and understanding the game of Set I’m excited to be doing two topics that get at spatial reasoning, as research indicates that girls especially can use more experience with this — and we’ll even have a tactile dimension with our Braille.

[A parent letter from November, 2012 at Emerson Elementary.] Dear Parents, A few notes from our recent and upcoming Math Circles: 1. In an effort to figure out how to teach girls better, I have hired a gender consultant, someone with an M.A. in Women and Gender Studies, to observe and critique my teaching, and to discuss with me some of the academic literature on gender and math. Sarah has visited class once so far, but she’s been great at spotting some of my teaching errors and challenging assumptions I have about working with girls. I would welcome more people to talk with about these issues. If any of you have a formal background in gender or simply an interest from having observed yourself and your children, please tell me and let’s talk. This is a moment when I am unusually teachable. One of the things that Sarah noticed last week was that three of the girls had independently written above their scratch work a sentence stating that the problem we were working on was hard. (It was, in fact it was impossible.) Reflecting on the fact that this dovetails with some of the literature on girl’s self-perception, I decided to begin class yesterday differently than I usually do. I started with three short anecdotes about my confronting, and sometimes failing to solve, problems in my own life that were hard. I then asked the children to reflect on when they’ve had problems that were really hard — in school or otherwise — and how they approached them. They became quiet and earnest. I told them that my teaching is not really about math; Math Circle could just as well be titled, “This is a Hard Problem. What are You Going to Do About It?” That’s as far as we got yesterday, but I’ll follow up in future lessons. 2. Yesterday we finished our penultimate lesson on parity (the mathematics of odd and even numbers). They’ve worked on five short problems that are each about parity , but which have had such varied settings that the children haven’t realized that the problems are all related. Next lesson will be a summation, or as I told them at the end yesterday, “Next lesson, all will be revealed — or not.” Sometimes in class we treat math as an inductive subject — the students try to find examples that satisfy a certain condition; when they can’t, they conclude, “It’s impossible.” This is fine as a heuristic, but in the end math needs to be deductive, depending upon proofs, and not only inductive, depending upon examples. The next lesson, I expect that they will do their first proof, and to make it more dramatic, we won’t fiddle with examples beforehand, I will invite them to dream up their parity proof out of pure thought. [Note: I did try this the following week, but it flopped miserably. That was probably my fault; most failures seem to happen when I get excited and rush the curriculum.] 3. Next problem due up: How much time would Santa Claus have to visit each family in America that he visits? How many reindeer would he need to pull his sleigh? This is a Fermi Problem, a back-of-the envelope calculation that uses a little bit of math and a lot of cleverness to estimate something in the real world. The classic Fermi Problem is, “How many piano tuners are there in Manhattan?” but that’s more of adult-interest. I made up the Santa Claus problem several years ago and it has been well received. At first I worried about its Christian reference, but everyone seems to get into the spirit of it. It should take three hours or so to muck through the problem.

[At Malcolm X, Math Circle is an optional lunchtime activity.] Dear Parents, At 11:39 I was bracing for the onslaught when a tiny girl, a first grader it seemed, came into our hallway on her way to the library. Only she wasn’t going to the library. “I’m here for Math Circle,” she announced to me. She had the quality of a quietly heralding angel. The photos attached are of the first ten minutes and the last 35 minutes. The first ten felt like a 60’s teach-in, full of the sincere, the curious, and those simply seeking to dine out in a new locale, 40+ in all. After a pleasant introduction to topology while they ate lunch, most then chose to go out to the playground, while sixteen chose to go to the library to work on the day’s challenge, figuring out which letters are topologically equivalent. They did this by classifying cut-outs of the letters or by playing a simple drawing game which I named “Morph-It.” The third photograph is two two girls at work playing Morph-It. Notice the second photograph: Math Circle classes usually attract more boys than girls, so it is thrilling to see so many girls here. I can only guess at why. If the alternative is to go out on the playground, perhaps many boys have a greater need than girls for what playground offers. Of the sixteen students, most girls came with one or two friends, but each boy, as far as I could tell, came as an individual. It may be that if girls can approach math with a friend, they are much less frightened or put off than if they face it alone. Lastly, I had prepared two lessons to teach, depending on the crowd, and I chose the one which is geometric rather than numerical, quietly profound rather than splashy and spectacular. That may have spoken more to the girls, or at least to a certain personality type.Whatever the reason for so many girls, that seems something precious that we should hold on to. As well, it was wonderful to meet several students whom I had not met before. There were several students whom I was expecting, but I am sorry did not show up, and others who stayed for the initial talk but chose not to continue the math challenge in the library. I saw one boy after school who had forgotten to come, and he felt bad for having missed it. It may take a couple of weeks, but we’ll get the kinks out. Topologically speaking, there are only nine distinct shapes that make up the letters of our alphabet. I heard afterwards from Mr. Hunt [the principal] that some students had come up to challenge him, how many letters are there in the alphabet? Then they explained to him that there are 9, not 26. Henri